Find the value of `'lambda'` such that the vectors `vec(a)` and `vec(b)` are perpendicular (orthogonal), where :
`vec(a)=7hat(i)-lambda hat(j)-7hat(k), vec(b)=4hat(i)+5hat(j)-hat(k)`

Answer

To find the value of \( \lambda \) such that the vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular (orthogonal), we can follow these steps:

### Step 1: Write down the vectors
The vectors are given as:
\[ \vec{a} = 7\hat{i} - \lambda\hat{j} - 7\hat{k} \]
\[ \vec{b} = 4\hat{i} + 5\hat{j} - \hat{k} \]

### Step 2: Use the condition for orthogonality
Two vectors are orthogonal if their dot product is zero:
\[ \vec{a} \cdot \vec{b} = 0 \]

### Step 3: Calculate the dot product
The dot product \( \vec{a} \cdot \vec{b} \) is calculated as follows:
\[ \vec{a} \cdot \vec{b} = (7\hat{i} - \lambda\hat{j} - 7\hat{k}) \cdot (4\hat{i} + 5\hat{j} - \hat{k}) \]
Using the distributive property of the dot product:
\[ = 7 \cdot 4 + (-\lambda) \cdot 5 + (-7) \cdot (-1) \]
Calculating each term:
\[ = 28 - 5\lambda + 7 \]
Combining the constants:
\[ = 35 - 5\lambda \]

### Step 4: Set the dot product to zero
To find \( \lambda \), we set the dot product equal to zero:
\[ 35 - 5\lambda = 0 \]

### Step 5: Solve for \( \lambda \)
Rearranging the equation gives:
\[ 5\lambda = 35 \]
Dividing both sides by 5:
\[ \lambda = 7 \]

### Conclusion
The value of \( \lambda \) such that the vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular is:
\[ \lambda = 7 \]

---